Under \(C_3\)  transformation, the spinor \(\Phi (x) \rightarrow e^{i \frac{2\pi }{3} \sigma _z} \Phi (x^{\prime })\) , where \(x^{\prime }\)  is the position \(x\)  rotated clockwise by \(2\pi /3\)  [1]}. Therefore, the two zero modes in Eq. REF  carry the extra phases of \(e^{i2\pi /3}\)  and \(e^{-i2\pi /3}\)  respectively, and thus transform differently under \(C_3\) . Furthermore, the chiral p-h transformation \(\sigma _z \otimes I\)  commutes with \(C_3\) . As the inter-layer tunneling \(T(x)\)  is gradually turned on [1]}, each of these two modes at \(K_m\)  must transform to itself under \(\sigma _z \otimes I\) , and therefore each still has zero energy. As a consequence, the two bands around the CNP touch at the Dirac cone at \(K_m\)  even when \(\mathcal {A}\)  is included.
